How can I make my own mesh and add it to Addmenu > Mesh?
You'd have to create your own add-on for this. First create a Python function that creates the mesh, then create an operator to call that Python function, then add the operator to a menu. There are loads of online tutorials on how to do these things; the Blender API documentation is a good start, as it also includes information about best practices and common pitfalls.

You can check out the built-in Add Extra Objects add-on to look at similar functionality.